arXiv subjects

Sarah Strikwerda

Publications and source records attributed to Sarah Strikwerda.

6 recordsLinked to original sources

arXiv2025

Sharp Interface Dynamics in a Minimal Non-Reciprocal Cahn-Hilliard System

Interest in non-reciprocally coupled systems recently led to the introduction of a minimal non-reciprocally coupled Cahn-Hilliard (CH) model by Brauns and Marchetti in 2024 arXiv:2306.08868, which we refer to as the Brauns-Marchetti (BM) model. This model can be seen as a conservative counterpart to the spatially extended FitzHugh-Nagumo model. Lacking a gradient structure, the BM model was observed to exhibit interesting dynamics including traveling periodic wave-trains and other coherent structures, as well as spatiotemporal chaos in certain parameter regimes. In this paper, we derive an effective equation for the interface dynamics of solutions to the BM model in $\mathbb{R}^2$ in the sharp-interface limit. The resulting system of equations is a generalization of the classical Mullins-Sekerka (MS) equations, which we refer to as the modified MS equations. We show that the modified MS equation shares some properties with its classical counterpart, but importantly, it is not in general a length minimizing flow. To illustrate the utility of this asymptotic reduction in the sharp interface limit, we perform a detailed analysis of stationary and periodic wave-trains, systematically deriving expressions for wave-train speeds and stability thresholds. The methods used here should be applicable to other non-reciprocally coupled CH models and therefore provide another avenue for their more detailed analysis.

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arXiv2025

A hierarchy of blood vessel models, Part I: 3D-1D to 1D

We propose and analyze a family of models describing blood perfusion through a tissue surrounding a thin blood vessel. Our goal is to rigorously establish convergence results among 3D-3D Darcy--Stokes, 3D-1D Darcy--Poiseuille, and 1D Green's function methods commonly used to model this process. In Part I, we propose a 3D-1D Darcy--Poiseuille system where the coupling across the permeable vessel surface involves an angle-averaged Neumann boundary condition coupled with a geometrically constrained Robin boundary condition. We show that this model is well-posed and moreover limits to a 1D Green's function model as the maximum vessel radius $\epsilon\to 0$. In the 1D model, the exterior blood pressure is given by an explicit Green's function expression involving the interior blood pressure. The interior pressure satisfies a novel 1D integrodifferential equation in which the integral term incorporates the effects of the exterior pressure and the vessel geometry. Much of this paper is devoted to analyzing this integrodifferential equation. Using the \emph{a priori} bounds obtained here, we show that the solution to the 1D model converges to the 3D-1D solution with a rate proportional to $\epsilon^{1/2}|\log\epsilon|$. In Part II [Ohm \& Strikwerda, arXiv preprint July 2025], we rely on the 1D estimates to show that both the 1D and 3D-1D models converge to a coupled 3D-3D Darcy-Stokes system as $\epsilon\to 0$, thereby establishing a convergence chain among all hierarchy levels.

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arXiv2025

A hierarchy of blood vessel models, Part II: 3D-3D to 3D-1D and 1D

We propose and analyze a hierarchy of three models of blood perfusion through a tissue surrounding a thin arteriole or venule. Our goal is to rigorously link 3D-3D Darcy--Stokes, 3D-1D Darcy--Poiseuille, and 1D Green's function methods commonly used to model this process. Here in Part II, we consider the most detailed level, a 3D-3D Darcy-Stokes system coupled across the permeable vessel surface by mass conservation and pressure/stress balance conditions. We derive a convergence result between the 3D-3D model and both the 3D-1D Darcy--Poiseuille model and 1D Green's function model proposed in Part I [Ohm \& Strikwerda, arXiv preprint July 2025] at a rate proportional to $\epsilon^{1/6}|\log\epsilon|$, where $\epsilon$ is the maximum vessel radius. The rate is limited by the inclusion of a degenerate endpoint where the vessel radius vanishes, i.e. becomes indistinguishable from a capillary. Key to our proof are \emph{a priori} estimates for the 1D integrodifferential model obtained in Part I.

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arXiv2025

Controlling Klein-Gordon Chains and Lattices

In this work, we initiate the study of controlling nonlinear Klein-Gordon chains and lattices through their emergent collective flocking behavior. By constructing appropriate feedback control mechanisms, we demonstrate that any physically admissible flock state can be achieved in finite time, meaning the chain can be driven from arbitrary initial vibrations toward a coherent traveling-wave motion. Finally, we reveal a deep connection between the flocking problem and a minimal-time control principle formulated within the framework of nonlinear Hamilton-Jacobi equations and optimal control theory, providing a unifying view-point for wave control in discrete nonlinear media.

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arXiv2022

Input Regularization for Integer Optimal Control in BV with Applications to Control of Poroelastic and Poroviscoelastic Systems

We revisit a class of integer optimal control problems for which a trust-region method has been proposed and analyzed in arXiv:2106.13453v3 [math.OC]. While the algorithm proposed in arXiv:2106.13453v3 [math.OC] successfully solves the class of optimization problems under consideration, its convergence analysis requires restrictive regularity assumptions. There are many examples of integer optimal control problems involving partial differential equations where these regularity assumptions are not satisfied. In this article we provide a way to bypass the restrictive regularity assumptions by introducing an additional partial regularization of the control inputs by means of mollification and proving a $\Gamma$-convergence-type result when the support parameter of the mollification is driven to zero. We highlight the applicability of this theory in the case of fluid flows through deformable porous media equations that arise in biomechanics. We show that the regularity assumptions are violated in the case of poro-visco-elastic systems, and thus one needs to use the regularization of the control input introduced in this article. Associated numerical results show that while the homotopy can help to find better objective values and points of lower instationarity, the practical performance of the algorithm without the input regularization may be on par with the homotopy.

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arXiv2021

Solving Bang-Bang Problems Using The Immersed Interface Method and Integer Programming

In this paper we study numerically solving optimal control problems with bang-bang control functions. We present a formal Lagrangian approach for solving the optimal control problem, and address difficulties encountered when numerically solving the state and adjoint equations by using the immersed interface method. We note that our numerical approach does not approximate the discontinuous control function with smooth functions, instead we solve the true bang-bang optimal control problem. Our approach for solving the optimal control problem uses an adjoint-based gradient. We use the gradient in our first-order trust-region method to generate a local minimizing control. We present detailed numerical results to demonstrate the effectiveness of our method.

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